Response Spectrum Analysis

Response spectrum analysis can be used to estimate the peak response (displacement, stress, etc.) of a structure to a particular base motion or force. The method is only approximate, but it is often a useful, inexpensive method for preliminary design studies.

The response spectrum procedure is based on using a subset of the modes of the system, which must first be extracted by using the eigenfrequency extraction procedure. The modes will include eigenmodes and, if activated in the eigenfrequency extraction step, residual modes. The number of modes extracted must be sufficient to model the dynamic response of the system adequately, which is a matter of judgment on your part.

If the number of eigenmodes included in the superposition does not sufficiently represent the total mass of the structure, you can use the missing mass method to augment the missing inertia in the dynamic response as described in Using the Missing Mass Method.

In cases with repeated eigenvalues and eigenvectors, the modal summation results must be interpreted with care. You should use mode combination rules that are appropriate for closely spaced modes.

While the response in the response spectrum procedure is for linear vibrations, the prior response may be nonlinear. Initial stress effects (stress stiffening) will be included in the response spectrum analysis if nonlinear geometric effects (General and Perturbation Procedures) were included in a general analysis step prior to the eigenfrequency extraction step.

The problem to be solved can be stated as follows: given a set of base motions, ${\ddot{u}}_j^B\left(t\right)$ ($j=1,2,3$), specified in orthogonal directions defined by direction cosines $t_k^j$ ($k=1,2,3$), estimate the peak value over all time of the response of any variable in a finite element model that is simultaneously subjected to these multiple base motions. The peak response is first computed independently for each direction of excitation for each natural mode of the system as a function of frequency and damping. These independent responses are then combined to create an estimate of the actual peak response of any variable chosen for output, as a function of frequency and damping.

The acceleration history (base motion) is not given directly in a response spectrum analysis; it must first be converted into a spectrum.

The response spectrum method is based on first finding the peak response to each base motion excitation of a one degree of freedom system that has a natural frequency equal to the frequency of interest. The single degree of freedom system is characterized by its undamped natural frequency, ${\omega }_{\alpha }$, and the fraction of critical damping present in the system, ${\xi }_{\alpha }$, at each mode $\alpha$. The equations of motion of the system are integrated through time to find peak values of relative displacement, relative velocity, and relative or absolute acceleration for the linear, one degree of freedom system. This process is repeated for all frequency and damping values in the range of interest. Plots of these responses are known as displacement, velocity, and acceleration spectra: $S^D\left({\omega }_{\alpha },{\xi }_{\alpha }\right)$, $S^V\left({\omega }_{\alpha },{\xi }_{\alpha }\right)$, and $S^A\left({\omega }_{\alpha },{\xi }_{\alpha }\right)$. The response spectrum can be obtained directly from measured data, as described in Defining a Spectrum Using Values of S as a Function of Frequency and Damping below. You can also use a Fortran program to define a spectrum; an example of defining a spectrum from an acceleration record in this way is provided in Analysis of a cantilever subject to earthquake motion.

Alternatively, you can create the required spectrum by specifying an amplitude (time history record), the frequency range, and the damping values for which the spectrum will be built, as described in Creating a Spectrum from a Given Time History Record below. The spectrum can be used in the subsequent response spectrum analysis, or it can be written to a file for future use.

For each damping value the magnitude of the response spectrum must be given over the entire range of frequencies needed, in ascending value of frequency. Abaqus/Standard interpolates linearly between the values given on a log-log scale. Outside the extremes of the frequency range given, the magnitude is assumed to be constant, corresponding to the end value given. (See Material Data Definition for an explanation of data interpolation.)

Any number of spectra can be defined, and each spectrum must be named. The response spectrum procedure allows up to three spectra to be applied simultaneously to the model in orthogonal physical directions defined by their direction cosines.

Since the response spectrum procedure uses modal methods to define a model's response, the value of any physical variable is defined from the amplitudes of the modal responses (the “generalized coordinates”), $q_{\alpha }$. The first stage in the response spectrum procedure is to estimate the peak values of these modal responses. For mode $\alpha$ and spectrum k this is

where

$q_{\alpha }$

is the modal amplitude for mode $\alpha$;

$c_k$

is a scaling parameter introduced as part of the response spectrum procedure definition for spectrum $S_k^D\left({\omega }_{\alpha },{\xi }_{\alpha }\right)$;

$S_k^D\left({\omega }_{\alpha },{\xi }_{\alpha }\right)$

is the user-defined value of the spectrum (see Specifying a Spectrum) in direction k interpolated, if necessary, at natural frequency ${\omega }_{\alpha }$ and the fraction of critical damping ${\xi }_{\alpha }$ in mode $\alpha$;

$t_j^k$

is the jth direction cosine for the kth spectrum; and

${\mathrm{\Gamma }}_{\alpha j}$

is the participation factor for mode $\alpha$ in direction j (see Natural Frequency Extraction).

Similar expressions for ${\left({\dot{q}}_{\alpha }^{max}\right)}_k$ and ${\left({\ddot{q}}_{\alpha }^{max}\right)}_k$ can be obtained by substituting velocity or acceleration spectra in the above equation.

The individual peak responses to the excitations in different directions will occur at different times and, therefore, must be combined into an overall peak response. Two combinations must be performed, and both introduce approximations into the results:

1. The multidirectional excitations must be combined into one overall response. This combination is controlled by the directional summation method, as described below in Directional Summation Methods.

2. The peak modal responses must be combined to estimate the peak physical response. This combination is controlled by the modal summation method, as described below in Modal Summation Methods.

Depending on the type of base excitation, either modal responses or directional responses are combined first.

Each spectrum can be divided into low-, medium-, and high-frequency regions. In the low-frequency region the modes are usually uncorrelated. In the mid-range they are partially correlated, and in the high-frequency region they tend to be correlated with the input data and, hence, between themselves. Each of these regions may, therefore, require special consideration. These considerations are addressed by the Gupta and Lindley-Yow methods.

We divide the modal response into two parts: the rigid part and the damped periodic part. It is assumed that the rigid part and the damped periodic part are statistically independent. Let us define the rigid response coefficient for each mode by $0\le {\alpha }_{\beta }\le 1$, the rigid response by ${\left(R_{\beta }^i\right)}^r{=\alpha }_{\beta }\left(R_{\beta }^i\right)$, and the periodic response by ${{\left(R_{\beta }^i\right)}^p=R_{\beta }^i\sqrt{1-{\alpha }_{\beta }^2}}^{}$. Therefore, the relationship between both regions becomes:

which is equivalent to summing both regions according to the SRSS rule. The value of the rigid response coefficient is chosen depending on the method used. For the Gupta method this coefficient is estimated based on two user-specified frequencies: $F_1$ defining the beginning of the periodic region, and $F_2$ defining the beginning of the rigid region (which is often equivalent to the frequency associated with zero period acceleration). For the current frequency $F_{\beta }$, this expression for the Gupta method is

For the Lindley-Yow method this coefficient is equal to

where ZPA denotes the zero period acceleration for the user-specified frequency value and $S_{a\beta }$ denotes the acceleration spectrum for mode $\beta$ . The Lindley-Yow method does not work well for the region of frequencies on the low-frequency end of the spectrum. Therefore, we set the coefficient ${\alpha }_{\beta }=0$ for all modes below the frequency of the peak acceleration value for this spectrum.

The rigid response method works with only two modal summation rules: CQC and DSC with their respective definitions for the mode correlation coefficients $c_{\alpha \beta }$. By default, no rigid response is included. To include a rigid response, you must specify the rigid response method.

RESPONSE SPECTRUM, COMP=comp, RIGID RESPONSE=GUPTA

RESPONSE SPECTRUM, COMP=comp, RIGID RESPONSE=LINDLEY-YOW

You can select the modes to be used in modal superposition and specify damping values for all selected modes.

You can use the missing mass method to augment the missing inertia in the dynamic response when the number of eigenmodes included in the superposition does not sufficiently represent the total mass of the structure. The missing inertia is due primarily to the truncated higher frequency modes that belong to the in-phase (rigid) response. Quasistatic residual modes in the direction of applied spectra can be substituted for these truncated modes. This technique is most suitable for use with the response spectra summations that split the frequency range into periodic and rigid phases, such as the Gupta or Lindley-Yow methods. However, any modal summation method can be used with the missing mass method. The missing mass method results in better response accuracy and is explained in three small spring-mass examples introduced first in the paper by Dhileep (Dhileep and Bose, 2008). The residual vector representing missing energy is found by a simple inertia balance in eigenspace:

where

$K$

represents the stiffness of the model,

$M$

represents the mass of the model,

${\mathrm{\Phi }}_i$ and ${\mathrm{\Gamma }}_i$

represent the $i^{th}$ out of $m$ eigenvectors and participation factors,

$u_B$

is the unit base displacement in the direction of the earthquake spectrum,

$a_g$

is a ground acceleration, and

$x_0$

is a pseudostatic residual mode representing the response of the remaining modes beyond the mode $m$.

RESPONSE SPECTRUM, MISSING MASS METHOD